Geometric Satake, Springer correspondence, and small representations II

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types.Comment: Version 4: minor revisions; 73 page

Modular perverse sheaves on flag varieties I: tilting and parity sheaves

In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.Comment: 42 pages; with a joint appendix with Geordie Williamso

Modular generalized Springer correspondence II: classical groups

We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-$0$ coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for $\mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $\mathrm{SO}(n)$ and $\mathrm{Sp}(2n)$ with characteristic-$2$ coefficients.Comment: 52 pages. Version 2 corrects a minor mistake in the combinatorics of the type D case; no numbered statements are affected. Version 3 has minor additions, mostly in Section 2; final version, to appear in J. Eur. Math. So

Modular generalized Springer correspondence: an overview

This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture `Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction.Comment: 19 pages. Version 2 includes more examples and tables in Section

Modular generalized Springer correspondence III: exceptional groups

We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$.Comment: 40 pages. Version 2: added section 7.5, modified Table 5.2 to match current conventions of GAP3. Version 3 has minor edits suggested by the referee, including a slight strengthening of Proposition 3.2; final version, to appear in Math. Annale

Rancang Bangun Pembelajaran Online Sistem Operasi Windows 7 dengan HTML 5

Website yang ada saat ini umumnya dibangun menggunakan tag – tag dalam HTML 4.1, dimana HTML 4.1 belum mendukung tampilan audio dan video, tetapi audio dan video dapat dilihat dan didengar dengan adanya aplikasi tambahan sesuai dengan keperluan. HTML 5 dikembangkan W3C yang telah mendukung audio dan video pada website tanpa harus tergantung pada aplikasi tambahan. Pada penulisan ini, penulis mengimplementasikan pembelajaran sistem operasi windows 7 di dalam website yang akan dibangun, dan penulis menggunakan metodologi waterfall, yang dimulai dengan tahap pendefinisian kebutuhan, analisis kebutuhan, perancangan sistem, pengkodean website, dan pengujian. Pada pengkodean website, penulis menggunakan actionscript untuk merancang konten dari website yang akan dibangun. Dari hasil rancang bangun website pembelajaran ini, diharapkan dapat membantu masyarakat untuk mengenal dan mengoperasikan sistem operasi windows 7 secara mendasar dengan mudah